Continuum models (whose Hamiltonians are polynomials in momentum,
typically describing the low-energy expansion of some more complicated, “more microscopic”
underlying Hamiltonian about the points of interest in the Brillouin zone)
are particularly appealing for theoretical studies of bound states, due to the relative simplicity of their bulk Hamiltonians.
The main challenge on this path is a systematic description of the boundary.
For continuum models, this description comes down to determining proper boundary conditions (BCs),
formulated as linear homogeneous relations between the wave-function components and their derivatives at the boundary.
It has been understood for quite some
time [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}
that BCs arise as the consequence of the fundamental principle of quantum mechanics:
norm conservation of the wave function during the time evolution described by the Schrödinger equation,
which leads to the hermiticity requirement for the Hamiltonian,
which in turn leads to the conservation of the probability current at the boundary.
This understanding naturally leads to the notion of general BCs,
as a family of all possible BCs (for a given Hamiltonian) that satisfy the current-conservation principle.
